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The main result is slightly more general than the following statement: Let f: X → Y be a quasi-perfect mapping, where X is a regular space and Y a Hausdorff totally non-meagre space; if X or Y is χ-scattered, or if Y is a Lasnev space, then X is totally non-meagre. In particular, the product of a compact space X and a Hausdorff regular totally non-meagre space Y which is χ-scattered or a Lasnev space, is totally non-meagre.

We show that for some large classes of topological spaces X and any metric space (Z,d), the point of continuity property of any function f: X → (Z,d) is equivalent to the following condition: (*) For every ε > 0, there is a neighbourhood assignment ${\left(V_x\right)}_{x\in X}$ of X such that d(f(x),f(y)) < ε whenever $(x,y)\in V_y\times V_x$. We also give various descriptions of the filters ℱ on the integers ℕ for which (*) is satisfied by the ℱ-limit of any sequence of continuous functions from a topological space into a metric space.

For a non-precompact topological group G, we consider the space C(G) of bounded, continuous, scalar-valued functions on G with the supremum norm, together with the subspace LMC(G) of left multiplicatively continuous functions, the subspace LUC(G) of left norm continuous functions, and the subspace WAP(G) of weakly almost periodic functions. We establish that the quotient space LUC(G)/WAP(G) contains a linear isometric copy of ${\ell }_{\infty }$, and that the quotient space C(G)/LMC(G) (and a fortiori C(G)/LUC(G))...